Complete lattices

 
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 \theoremstyle{definition}
 \newtheorem{definition}{Definition}
 \newtheorem*{morphisms}{Morphisms}
 \newtheorem*{basic_results}{Basic Results}
 \newtheorem*{examples}{Examples}
 \newtheorem{example}{}
 \newtheorem*{properties}{Properties}
 \newtheorem*{finite_members}{Finite Members}
 \newtheorem*{subclasses}{Subclasses}
 \newtheorem*{superclasses}{Superclasses}
 \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
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 \begin{document}
 \textbf{\Large Complete lattices}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Complete_lattices}{edit}
 
 \abbreviation{CLat}
 \begin{definition}
 A \emph{complete lattice} is a structure $\mathbf{L}=\langle L,\bigvee,\bigwedge\rangle$ such that $\bigvee,\bigwedge$ map
 subsets of $L$ to elements of $L$ and
 
 
 $\langle L,\vee,\wedge\rangle$ is a \href{Lattices.pdf}{Lattices}
 
 
 $\bigvee S$ is the least upper bound of $S$
 
 
 $\bigwedge S$ is the greatest lower bound of $S$
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{L}$ and $\mathbf{M}$ be complete lattices. 
 A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a complete homomorphism: 
 
 $h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$
 \end{morphisms}
 \begin{basic_results}
 \end{basic_results}
 \begin{examples}
 \begin{example}
 $\langle \mathcal{P}(X),\bigcup,\bigcap\rangle$, the set of all subsets of a set $X$, with union and intersection of families of sets.
 \end{example}
 \end{examples}
 
 \begin{table}[h]
 \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
 
 \begin{tabular}{|ll|}\hline
 Classtype & Second-order\\\hline
 Amalgamation property & Yes\\\hline
 Strong amalgamation property & Yes\\\hline
 Epimorphisms are surjective & Yes\\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Algebraic_lattices.pdf}{Algebraic lattices} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Lattices.pdf}{Lattices} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %

 
 \documentclass[12pt]{amsart}
 \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
 \parindent=0pt
 \parskip=5pt
 \addtolength{\oddsidemargin}{-.5in}
 \addtolength{\evensidemargin}{-.5in}
 \addtolength{\textwidth}{1in}
 \theoremstyle{definition}
 \newtheorem{definition}{Definition}
 \newtheorem*{morphisms}{Morphisms}
 \newtheorem*{basic_results}{Basic Results}
 \newtheorem*{examples}{Examples}
 \newtheorem{example}{}
 \newtheorem*{properties}{Properties}
 \newtheorem*{finite_members}{Finite Members}
 \newtheorem*{subclasses}{Subclasses}
 \newtheorem*{superclasses}{Superclasses}
 \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
 \pagestyle{myheadings}\thispagestyle{myheadings}
 \markboth{\today}{math.chapman.edu/structures}
 
 \begin{document}
 \textbf{\Large Complete lattices}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Complete_lattices}{edit}
 
 \abbreviation{CLat}
 \begin{definition}
 A \emph{complete lattice} is a structure $\mathbf{L}=\langle L,\bigvee,\bigwedge\rangle$ such that $\bigvee,\bigwedge$ map
 subsets of $L$ to elements of $L$ and
 
 
 $\langle L,\vee,\wedge\rangle$ is a \href{Lattices.pdf}{Lattices}
 
 
 $\bigvee S$ is the least upper bound of $S$
 
 
 $\bigwedge S$ is the greatest lower bound of $S$
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{L}$ and $\mathbf{M}$ be complete lattices. 
 A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a complete homomorphism: 
 
 $h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$
 \end{morphisms}
 \begin{basic_results}
 \end{basic_results}
 \begin{examples}
 \begin{example}
 $\langle \mathcal{P}(X),\bigcup,\bigcap\rangle$, the set of all subsets of a set $X$, with union and intersection of families of sets.
 \end{example}
 \end{examples}
 
 \begin{table}[h]
 \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
 
 \begin{tabular}{|ll|}\hline
 Classtype & Second-order\\\hline
 Amalgamation property & Yes\\\hline
 Strong amalgamation property & Yes\\\hline
 Epimorphisms are surjective & Yes\\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Algebraic_lattices.pdf}{Algebraic lattices} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Lattices.pdf}{Lattices} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %

http://mathcs.chapman.edu/structuresold/files/Complete_lattices.pdf

%%run pdflatex

%

\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\addtolength{\oddsidemargin}{-.5in}
\addtolength{\evensidemargin}{-.5in}
\addtolength{\textwidth}{1in}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\pagestyle{myheadings}\thispagestyle{myheadings}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Complete lattices}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Complete_lattices}{edit}

\abbreviation{CLat}
\begin{definition}
A \emph{complete lattice} is a structure $\mathbf{L}=\langle L,\bigvee,\bigwedge\rangle$ such that $\bigvee,\bigwedge$ map
subsets of $L$ to elements of $L$ and


$\langle L,\vee,\wedge\rangle$ is a \href{Lattices.pdf}{Lattices}


$\bigvee S$ is the least upper bound of $S$


$\bigwedge S$ is the greatest lower bound of $S$
\end{definition}
\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be complete lattices. 
A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a complete homomorphism: 

$h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$
\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\langle \mathcal{P}(X),\bigcup,\bigcap\rangle$, the set of all subsets of a set $X$, with union and intersection of families of sets.
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & Second-order\\\hline
Amalgamation property & Yes\\\hline
Strong amalgamation property & Yes\\\hline
Epimorphisms are surjective & Yes\\\hline
\end{tabular}
\end{properties}
\end{table}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\ 

\href{Algebraic_lattices.pdf}{Algebraic lattices} 

\end{subclasses}
\begin{superclasses}\ 

\href{Lattices.pdf}{Lattices} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%